A man covers half of his journey by train at 60 km/h, half of the remaining distance by bus at 30 km/h and the rest by cycle at 10 km/h. What is his average speed (in km/h) for the entire journey?

Introduction / Context:
This question tests your understanding of average speed when a journey is divided into segments with different speeds and different fractions of the total distance. The key is to handle distance fractions carefully and compute the total time taken before finding the average speed over the entire journey.

Given Data / Assumptions:

- Total journey distance = D km (unknown).
- First half of the journey (D / 2) is covered by train at 60 km/h.
- Half of the remaining distance, that is D / 4, is covered by bus at 30 km/h.
- The final D / 4 is covered by cycle at 10 km/h.
- All speeds are constant for their respective segments.

Concept / Approach:
Average speed over an entire trip is defined as total distance divided by total time. Here the total distance is D, so we need to compute the time taken on each segment and add them. Once the total time is known, average speed is simply D divided by that total time. The absolute value of D will cancel out in the computation, so we can leave it symbolic.

Step-by-Step Solution:
First segment distance = D / 2 at 60 km/h. Time for first segment = (D / 2) / 60 = D / 120 hours. Remaining distance after first segment = D / 2. Second segment distance = half of the remaining = D / 4 at 30 km/h. Time for second segment = (D / 4) / 30 = D / 120 hours. Third segment distance = remaining D / 4 at 10 km/h. Time for third segment = (D / 4) / 10 = D / 40 hours. Total time T = D / 120 + D / 120 + D / 40. Combine: D / 120 + D / 120 = D / 60. So T = D / 60 + D / 40. Take common denominator 120: T = 2D / 120 + 3D / 120 = 5D / 120 = D / 24 hours. Average speed = total distance / total time = D / (D / 24) = 24 km/h.

Verification / Alternative check:
We can choose a convenient value for D, for example D = 24 km. Then time by train = 12 / 60 = 0.2 hours, time by bus = 6 / 30 = 0.2 hours, time by cycle = 6 / 10 = 0.6 hours. Total time = 1.0 hour for 24 km, giving an average speed of 24 km/h. This numeric check matches the symbolic solution exactly.

Why Other Options Are Wrong:
32 km/h, 20 km/h, and 18 km/h: These values do not result from the actual distance fractions and speeds. They could come from incorrectly averaging the speeds directly or misunderstanding the phrasing “half of the remaining distance.” Average speed must be based on total distance divided by total time, not a simple arithmetic mean of the speeds.

Common Pitfalls:
Many learners incorrectly take the average of 60, 30, and 10, or they treat each third of the distance equally instead of using half and quarter segments. Another error is to forget that the total distance is split as D / 2, D / 4, and D / 4. Always write each segment's distance and time clearly before computing average speed.

Final Answer:
The average speed over the entire journey is 24 km/h.